i don't have the mel at hand but i wouldn't be surprised if this is a no-go item... What you're actually checking is, that in the event of both boost pumps failure and an additional main ejector failure the check valve cuts out the fuel supply line from remaining main ejector to the failed main ejectors engine, because the fuel supply to both engines cannot be guaranteed by one main ejector alone. So one engine supply is cut out, which leaves you single-engine in flight but at least with full fuel pressure on the remaining good engine.
Concerning the first flight of the day, this is how we do it in our airline: We start engine no 1, then after it's stabilized we switch off both fuel pumps and wait for the valve to close (might take 10-15 seconds) and the eicas message (in addition to the fuel pump messages) RIGHT FUEL LO PRESS to appear. Then before engine shutdown after the first flight, we shut off engine no 1 first and switch off both fuel pumps and wait for LEFT FUEL LO PRESS to appear on the eicas.
Hope this helps a bit